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Mahler’s measure and special values of \(L\)-functions. (English) Zbl 0932.11069
The logarithmic Mahler’s measure of a complex polynomial \(P(x_1,\dots,x_n)\) is defined by the formula \[ m(P)=\int_0^1\cdots\int_0^1\log(|P(e(t_1),\dots,e(t_n)|) \,dt_1\cdots dt_n, \] where \(e(t)=\exp(2\pi it)\). It has been observed that for certain polynomials this measure is related to Dirichlet \(L\)-functions [see G. A. Ray, Can. J. Math. 39, 694–732 (1987; Zbl 0621.12005)] or \(L\)-functions of elliptic curves. Recently C. Deninger [J. Am. Math. Soc. 10, 259–281 (1997; Zbl 0913.11027)] proved that in certain cases relations of the form \(m(P)=rL'(E,0)\), where \(E\) is an elliptic curve and \(r\) is a rational number, are consequences of the Bloch-Beilinson conjectures concerning special values of \(L\)-functions. This applies in particular to the equality \(m(P)=L'(E,0)\), where \(P\) is the Laurent polynomial \(x+1/x+y+1/y+1\) and \(E\) is the elliptic curve of conductor 15. Numerically both sides of this equality agree on 50 decimal places.
In this paper the author describes the results of an extensive numerical search which lead to several other examples of polynomials \(P\) with the property that the ratio \(m(P)/L'(E,0)\) (where \(E\) is a suitable elliptic curve) is extremely close to a rational number (and conjectures that in fact one has equality here). He formulates conjectural sufficient conditions for this behaviour and notes that under additional assumptions the sufficiency of these conditions has been shown by H. C. Bornhorn [Preprint 35, Math. Inst. Univ. Münster, January 1999, see also his thesis (1998; Zbl 1109.11315)] and F. Rodriguez-Villegas [Topics in number theory. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 467, 17–48 (1999; Zbl 0980.11026)] to be a consequence of the Bloch-Beilinson conjectures.

MSC:
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11Y99 Computational number theory
Software:
PARI/GP; ecdata
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